A

Step 1: Given the acceleration of the particle as $a = kt$.
Step 2: We know that acceleration is the derivative of velocity with respect to time, so:
$$ a = \frac{dv}{dt} = kt $$
Step 3: To find the expression for velocity, we integrate the equation:
$$ \int dv = \int kt \, dt $$
Step 4: This gives:
$$ v = \frac{kt^2}{2} + C $$
Step 5: Since the motion starts from rest, the initial velocity $v(0) = 0$.
Step 6: Therefore, the constant of integration $C = 0$.
So, we have:
$$ v = \frac{kt^2}{2} $$
Thus, the final expression for velocity in terms of time $t$ is $v = \frac{kt^2}{2}$.
Therefore, the correct answer is A.